NOTES ON COBORDISM THEORY by Robert E. Stong Mathematical Notes, Princeton University Press 1968

A Detailed Table of Contents compiled by Peter Landweber and Doug Ravenel in November, 2007 based on decades of careful reading

Chapter I. Introduction–Cobordism Categories...... 1 Cobordism categories ...... 3 Relative cobordism...... 9

Chapter II. Manifolds with Structure – the Pontrjagin-Thom theorem ...... 14 (B, f) structures...... 14–16 Generalized Pontrjagin-Thom theorem ...... 18–23 Tangential structures, sequences of maps, ring structure, relative groups ...... 23–26

Chapter III. Characteristic Classes and Numbers ...... 27 Spectra...... 27 Ring spectra...... 28 Thom class...... 29 Fundamental class ...... 30 Characteristic class, characteristic number ...... 31 Orientation and Thom isomorphism ...... 36 Atiyah duality...... 37 Alexander and Spanier-Whitehead duality...... 39

Chapter IV. The Interesting Examples – A Survey of the Literature ...... 40 fr Example 1: Framed cobordism Ω∗ ...... 40 Example 2: Unoriented cobordism N∗ ...... 40 U Example 3: Complex cobordism Ω∗ ...... 41 SO Example 4: Oriented cobordism Ω∗ ...... 42 Example 5: w1 spherical cobordism W∗ ...... 43 Example 6: Bordism Ω∗(B, f)[X,A]...... 43 SU Example 7: Special unitary cobordism Ω∗ ...... 44 U Example 8: c1 spherical cobordism W∗ ...... 45 Spin Example 9: Spin cobordism Ω ...... 46 ∗ c c Spin Example 10: Spin cobordism Ω∗ , etc...... 47 c-s Example 11: Complex-Spin cobordism Ω∗ ...... 48 Sp Example 12: Symplectic cobordism Ω∗ ...... 48 Fifteen more examples and two pseudoexamples ...... 48–58

Chapter V. Cohomology of Classifying Spaces ...... 59 Vector bundles...... 59 Definition of characteristic classes ...... 61 Splitting lemma...... 65 Thom spaces...... 66 Ordinary cohomology of Grassmannians...... 69 Relationship between fields...... 73 Characteristic numbers of manifolds (projective spaces, Milnor hypersurfaces) ...... 75 Cohomology of BO and BSO ...... 81 Pontrjagin classes...... 82 Euler class...... 85

Chapter VI. Unoriented Cobordism ...... 90 The mod 2 Steenrod algebra A2 ...... 91 Adem relations ...... 91 Cartan formula ...... 92 Structure theorem for N∗ ...... 96 Wu classes vk, v(M)...... 98–100 Wu relations on characteristic numbers ...... 100–101 Relation to framed cobordism: the Hopf invariant ...... 102 Unoriented bordism: Steenrod representation ...... 106

Chapter VII. Complex Cobordism ...... 110 The mod p Steenrod algebra Ap ...... 111 U Structure of Ω∗ ...... 117 Complex K-theory ...... 117 Chern character ...... 117 Calculation of K-theory characteristic numbers...... 119 Construction of almost complex manifolds with certain characteristic numbers...... 125 U Ω∗ is polynomial ...... 128 U Polynomial generators for Ω∗ ...... 128 Relations among characteristic numbers [Stong-Hattori theorem] ...... 129

Relation to framed cobordism: the Adams unvariant eC ...... 133 Relation to unoriented cobordism ...... 137 Complex bordism ...... 143

Chapter VIII. σ1-Restricted Cobordism ...... 147 det(µ), µ an n-plane bundle ...... 147 P (Kr)-structure, K = R or C...... 147 W∗(K, r)...... 148 Semi-geometric methods: W∗(K, 2)...... 151 SG Relation between W∗(K, 2) and Ω∗ : Semi-geometric methods ...... 168 Relation to bordism groups ...... 172

Chapter IX. Oriented Cobordism ...... 176 SO Structure of Ω∗ ⊗ Q ...... 177 SO Torsion in Ω∗ is 2-primary ...... 180 U SO Ω∗ → Ω∗ /Torsion is onto ...... 180 SO Ω∗ /Torsion is polynomial ...... 180 SO Polynomial generators of Ω∗ /Torsion...... 180 SO All torsion in Ω∗ has order 2...... 182 SO Pontrjagin and Stiefel-Whitney numbers determine classes in Ω∗ ...... 183 SO Image of Ω∗ → N∗ ...... 185 Integrality theorem for oriented manifolds...... 200 Hirzebruch L class L(ξ)...... 202 Relations among Pontrjagin numbers ...... 207 The Ab class...... 208 Oriented bordism ...... 209 Relation to framed cobordism...... 213 The Pontrjagin numbers of an oriented manifold with framed boundary ...... 215 Relation to unoriented cobordism ...... 216 Relation to complex cobordism ...... 218 The index (or signature) ...... 219 The Hirzebruch index (or signature) theorem...... 223 Odd primary data ...... 224 Two primary data ...... 229 ∗ H (MSO; Z2) as a module over A2 ...... 233

Chapter X. Special Unitary Cobordism ...... 237 SU Structure of Ω∗ ⊗ Q ...... 238 SU Torsion in Ω∗ is 2-primary ...... 239 Construction of SU-mainfolds with certain characteristic numbers ...... 239 SU 1 Ω∗ ⊗ Z[ 2 ] is polynomial...... 242 SU All torsion in Ω∗ has order 2...... 243 SU Torsion in Ω∗ ...... 248 KO-theory characteristic numbers...... 249 Chern numbers of SU-manifolds...... 255 SU Ω∗ is determined by integral cohomology and KO characteristic numbers ...... 261 Product in W∗(C, 2)...... 262 Relation to framed cobordism...... 267 Relation to complex cobordism ...... 273 Relation to unoriented cobordism ...... 276 Relation to oriented cobordism...... 278

Chapter XI. Spin, Spinc, and Similar Nonsense ...... 283 Clifford algebra Cliff(V )...... 285 Spin(k), Spinc(k)...... 287 Pin(k), Pinc(k)...... 289 ∗ H (BSpin; Z2)...... 292 Connective covers of BO and BU ...... 295 Filtration of KO∗(X) and K∗(X)...... 302 Isomorphic homologies ...... 308 2-primary analysis of MSpin and MSpinc ...... 319 Spin Spinc Structure of Ω∗ , Ω∗ ...... 336 Spin KO-theory and mod 2 cohomology characteristic numbers determine Ω∗ ...... 337 Spinc Ordinary (Q, Z2) cohomology characteristic numbers determine Ω∗ ...... 337 Spin Basis for Ω ⊗ ...... 339 ∗ c Z2 U Spin Ω∗ → Ω∗ /Torsion is onto ...... 348 Relation to framed cobordism...... 350 Relation to unoriented cobordism ...... 350 Relation to oriented cobordism...... 351 Relation to complex cobordism ...... 353 Relation of Spin and Spinc ...... 354

Appendix 1. Advanced Calculus (23 pages)

Appendix 2. Differential Topology (25 pages)

Bibliography (142 items, 8 pages)